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# Mathematics

In science, there are no such things as unrelated courses. One finds that a mathematical form that represents competition in an ecology course stands a good chance of being the same one that represents a chemical reaction or a knight’s move on a chess board. This is one of the beauties of mathematics and, at the same time, one of its powers. Mathematics is a field that (by its own structure) uses forms to investigate itself. This power of introspection will become more and more manifest as you move up the course ladder. Indeed, learning mathematics at Marlboro is learning to spot the reflections of one course in another and to see that there is a great unity and beauty to the subject.

The mathematics curriculum at Marlboro has a two-fold purpose:

**Power:**to exercise the ability to formalize and abstract, and to develop a facility in specializing abstractions in order to attack applied problems.**Introspection:**to appreciate the inherent beauty of mathematical enquiry, and to investigate and to understand the intertwined relationships between the three major branches of mathematics.

Mathematics courses at Marlboro are designed to serve both of these goals, though some are skewed strongly in favor of one or the other.

My personal research is in the areas of combinatorics and group theory, especially in questions motivated by the design of experiments. These areas are very accessible to study at the undergraduate level, and expose students to both the problem solving and theory building sides of mathematics.

### Areas of Interest for Plan-level Work:

Students are encouraged to pursue whatever topic demands their attention from across the many sub-fields of mathematics. Recent examples include chaos theory, combinatorics, complex analysis, cryptography, formal languages, game theory, graph theory, group theory, history of mathematics, mathematical modeling and statistics. Interdisciplinary Plans are welcomed.

### Starting Points (Basic and Introductory Courses)

**TOPICS IN ALGEBRA, TRIGONOMETRY & PRE-CALCULUS** (*NSC556*)

This course covers a wide range of math topics prerequisite for further study in mathematics and science and of interest in their own right. The course is divided into over 50 units (listed on the course web page). One credit will be earned for each group of 6 units completed. Students select units to improve their weak areas. There are also tailored streams for students who wish to go on to study calculus or statistics and for those who wish to prepare for the GRE exam. Over this semester and next, 42 units will be offered in the timetabled sessions. Individual tutorial-style arrangements can be made to study the non-timetabled units or to study units earlier than their scheduled session. Prerequisite: None *Introductory | Credits: Variable*

**ASPECTS OF GEOMETRY** (*NSC563*)

Throughout the history of geometry, great advances have been made through radical reconceptualizations of the entire subject: Euclid’s axiomatic geometry; the analytic geometry of Descartes, et. al.; the projective geometry stemming from Renaissance art; the unification of geometry and number through complex numbers, quaternions and linear algebra; the discovery of non-Euclidean geometries by Gauss, Lobachevsky and Bolyai; the group theoretical synthesis of geometry of Felix Klein’s 1872 Erlanger Programm. We will focus on conceptual aspects of these points of view and see how each shift addressed fundamental issues left unresolved by existing theories. Prerequisite: None *Introductory | Credits: Variable*

**CALCULUS** (*NSC515*)

A one semester course covering differential and integral calculus and their applications. This course provides a general background for more advanced study in mathematics and science. Prerequisite: Topics in Algebra, Trigonometry and Pre-Calculus or equivalent *Introductory | Credits: 4*

**DISCRETE MATHEMATICS** (*NSC562*)

Discrete math is the study of mathematical objects on which there is no natural notion of continuity. Examples include the integers, networks, permutations and search trees. After an introduction to the tools needed to study the subject, the emphasis will be on you *doing* mathematics. Series of problems will lead gradually to proofs of major theorems in various areas of the discipline. This course is recommended for those intending to do advanced work in math or computer science. Prerequisite: None *Introductory | Credits: 4*

**STATISTICS** (*NSC123*)

We look at three main topics: the collection and presentation of data, the probability theory behind statistical methods and the analysis of data. Statistical tests covered include the t-test, linear regression, ANOVA and chi-squared. The open source statistical computing package R is introduced and used throughout the class. Prerequisite: Topics in Algebra, Trigonometry and Pre-Calculus or equivalent *Introductory | Credits: 4*

**A WHIRLWIND TOUR OF MATHEMATICS** (*NSC577*)

Do you want a thorough understanding of the most important and deep theorems in every branch of mathematics? Do you want to achieve this in a three credit course from a standing start? Good luck with that—you won’t manage it in this course. Instead, we’ll look at six to ten topics, chosen for their accessibility and beauty, and drawn from a broad range of sub-disciplines of math. The interests of students in the class will drive the exact choice of topics. Possibilities include: irrational and imaginary numbers, the infinite, chaos and fractals, Fermat’s Last Theorem, the Platonic solids, the fourth dimension, the combinatorial explosion, P vs. NP, the Four Color Theorem, non-Euclidean geometry, logical paradoxes and many others. No prior mathematical experience is expected. Prerequisite: None *Introductory | Credits: 3*

**GROUP THEORY & RUBIK’S CUBE** (*NSC203*)

This course will explain the basics of a branch of mathematics called group theory by examining Rubik’s Cube and other similar puzzles. My hope is that the puzzles will motivate the ideas behind group theory. Although this is an introductory course and does not depend on any previous math (we will, for example, hardly use numbers at all), students should be comfortable with abstract thought. Prerequisite: Some facility with abstract concepts *Introductory | Credits: 2*

**PUZZLED?** (*NSC541*)

This course will give students a chance to test and develop their puzzle-solving ingenuity. We’ll attack a series of puzzles, going from Lewis Carroll’s logic problems via the classic “recreational math” puzzles of Lucas, Loyd and Dudeney to modern crazes such as the sudoku. Pass/Fail grading. Prerequisite: None *Introductory | Credits: 2*

### Pursuing Interests (Intermediate and Thematic Courses)

**CALCULUS II** (*NSC212*)

We build on the theory and techniques developed in Calculus (NSC515). Topics include techniques and applications of integration, epsilon/delta definitions, power series, parametric equations and differential equations. Prerequisite: Calculus or permission of the instructor *Intermediate | Credits: 4*

**EXCURSIONS IN CALCULUS** (*NSC576*)

A deeper study and appreciation of the ideas of calculus, focusing on intuitive understanding of key concepts as well as the contemporary and historical role of the calculus in mathematics and science as conveyed by its wealth of important and beautiful applications. Prerequisite: Calculus or equivalent, or concurrent enrollment in Calculus *Introductory | Credits: 2*

**GAME THEORY** (*CDS31*)

While game theory is a field of mathematics, little math will be used in this course. The intent of this course is to provide students with a formal method for looking at the interactions of individuals (either firms, states, people or bacteria) and explaining or prediction the outcome. Optimally, students will learn or find applications for game theory in their respective fields. Prerequisite: None *Introductory | Credits: 3*

**ALGEBRAIC STRUCTURES** (*NSC524*)

An investigation of the properties of groups, rings, fields and vector spaces. Prerequisites: Permission of instructor, facility with vectors and matrices, several math courses *Intermediate | Credits: Variable*

**DIFFERENTIAL EQUATIONS** (*NSC384*)

Learn how to solve differential equations in closed form and by approximation when closed form is impossible or impractical. Applications will cover topics from fields of interest of students enrolled in the course. We will follow a text, but the largest part of the course will concentrate on computer simulations and solutions. Prerequisite: Concurrent enrollment in Calculus II *Intermediate | Credits: 4*

**LINEAR ALGEBRA** (*NSC164*)

Next to Calculus, this is the most important math course offered. It is important for its remarkable demonstration of abstraction and idealization on the one hand, and for its applications to many branches of math and science on the other. Whereas Calculus introduces undergraduates to a large warehouse of constantly used mathematical items, Linear Algebra has the power to use and manipulate those items. Matrices, vector spaces and transformations are studied extensively (most work is done in the n-dimensional real case). Prerequisite: Calculus *Intermediate | Credits: 4*

**NUMBER THEORY** (*NSC514*)

Numbers have been a source of fascination since ancient times. We investigate some of the more intriguing properties numbers can have, and study the work of some of the great mathematicians, including Euclid, Fermat and Gauss. We also look at cryptography—a modern application of number theory. Prerequisite: Topics in Algebra, Trigonometry and Pre-Calculus or equivalent *Intermediate | Credits: 4*

**STATISTICS WORKSHOP** (*NSC574*)

A follow-up to Statistics (NSC123) in which students will acquire and hone the statistical skills needed for their work on Plan. Course content will be driven by the interests and requirements of those taking the class. Prerequisite: Statistics (NSC123) or permission of the instructor *Intermediate | Credits: Variable*

**MULTIVARIABLE CALCULUS **(*NSC603*)

An extension of the ideas from Calculus and Calculus II to multivariable and vector functions. Topics covered include the geometry of 3-dimensional space, partial derivatives, multiple integrals and higher dimensional analogues of the fundamental theorem of calculus. Prerequisite: Calculus II or equivalent*Intermediate | Credits: 4*

**PROBABILITY **(*NSC604*)

Probabilities pop up every day like “There’s a 30 percent chance of rain” or “The probability of being dealt a full house in stud poker is approximately 0.00144.” Our main goal for the class will be developing various tools to calculate probabilities. Topics include axioms of probability, counting techniques, conditional probability, discrete and continuous random variables, special discrete and continuous distributions and joint distributions. Prerequisite: Calculus I *Intermediate | Credits: 4*

**DYNAMICAL SYSTEMS AND CHAOS THEORY **(*NSC605*)

Our goal for this class will be to study dynamical systems. To study dynamical systems, we will be using the software Mathematica (no initial computer programming knowledge is required). We will not only learn some of the theory behind dynamical systems, but we will also experiment on the computer. We will look at simple dynamical systems, use graphical analysis to help describe the behavior of a system, explore symbolic dynamics, examine fractals and look at the Mandelbrot set and Julia sets. Prerequisite: Linear Algebra or instructor's approval *Intermediate | Credits: 4*

**ADVANCED CALCULUS** (*NSC207*)

The geometry of Euclidean space and generalizations to n-dimensions. The study of differentiation and integration for functions of several variables, coordinate changes and the geometry of maps between spaces. Double and triple integration. Prerequisite: Permission of instructor *Advanced | Credits: 4*

**COMPLEX VARIABLES** (*NSC590*)

An introduction to functions of one complex variable. We look at geometry and transformations in the complex plane and then extend the notions of calculus of functions of real variables to this setting. Prerequisite: Calculus II *Advanced | Credits: 4*

**REAL ANALYSIS** (*NSC336*)

Real analysis is the study of the real numbers and functions of real numbers. After looking in some detail at the underpinnings of the real number system we’ll consider sequences, continuity, differentiation and integration. This course will contain very few theorems that you haven’t seen and used in Calculus. However, our interest here will be on their proofs rather than their applications. Prerequisites: Calculus and permission of instructor *Advanced | Credits: 4*

**WRITING MATH** (*NSC534*)

In this class you will study the writing and presentation of mathematics. All skills needed for writing Plan-level math will be discussed, from the overall flow of a paper to the use of the typesetting package LaTeX, via writing proofs well and choosing good examples. You will write short papers, based on material in your other math classes, that we will read and discuss as a group. May be repeated for credit. Prerequisite: Permission of instructor and concurrent math class *Advanced | Credits: 1*

### Good Foundation for Plan

All students of mathematics must search for a basic understanding in each of the three major areas: algebra, geometry and analysis. To accomplish this in one undergraduate lifetime, it is important to get some core courses—Discrete Math, Calculus, Calculus II and Linear Algebra—completed as soon as possible. Doing this will widen your range of options as you look for a topic on which to focus. When pursuing an interdisciplinary Plan, there is less urgency to complete these courses, since the “introspection” side of the Plan will turn towards investigating the relationships between mathematics and the chosen field.

### Sample Tutorial Topics

- Group Theory
- Statistical and Combinatorial Design Theory
- Cryptography
- Topology
- Combinatorics and Graph Theory
- Lie Theory
- Foundations of Mathematics